IS/ISO 11843-4 : 2003 Hkkjrh; ekud lalwpu l{kerk Hkkx 4 fn, x, eku ls U;wure lalwpd dh rqyuk djus dh iz.kkyh Indian Standard CAPABILITY OF DETECTION PART 4 METHODOLOGY FOR COMPARING THE MINIMUM DETECTABLE VALUE WITH A GIVEN VALUE ICS 03.120.30; 17.020 © BIS 2010 B U R E AU O F I N D I A N S TA N DA R D S MANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG NEW DELHI 110002 January 2010 Price Group 4 Statistical Methods for Quality and Reliability Sectional Committee, MSD 3 NATIONAL FOREWORD This Indian Standard (Part 4) which is identical with ISO 11843-4 : 2003 `Capability of detection -- Part 4: Methodology for comparing the minimum detectable value with a given value' issued by the International Organization for Standarization (ISO) was adopted by the Bureau of Indian Standards on the recommendation of the Statistical Methods for Quality and Reliability Sectional Committee and approval of the Management and Systems Division Council. An ideal requirement for the capability of detection with respect to a selected state variable would be that the actual state of every observed system can be classified with certainty as either equal to or different from its basic state. However, due to systematic and random distortions, this ideal requirement cannot be satisfied because: a) in reality all reference states, including the basic state, are never known in terms of the state variable. Hence, all states can only be characterized correctly in terms of differences from basic state, that is, in terms of the net state variable; b) in order to prevent erroneous decisions, it is generally recommended to report differences from the basic state only, that is, data in terms of the net state variable; and c) furthermore, the calibrations and the processes of sampling and sample preparation add random variation to the measurement results. In this standard, -- the probability is of detecting (erroneously) that a system is not in the basic state when it is in the basic state; and -- the probability is of (erroneously) not detecting that a system, for which the value of the net state variable is equal to the minimum detectable value (xd), is not in the basic state. The text of ISO Standard has been approved as suitable for publication as an Indian Standard without deviations. Certain conventions are, however, not identical to those used in Indian Standards. Attention is particularly drawn to the following: a) Wherever the words `International Standard' appear referring to this standard, they should be read as `Indian Standard'. b) Comma (,) has been used as a decimal marker in the International Standard while in Indian Standards, the current practice is to use a point (.) as the decimal marker. In this adopted standard, reference appears to certain International Standards for which Indian Standards also exist. The corresponding Indian Standards which are to be substituted in their respective places are listed below along with their degree of equivalence for the edition indicated: International Standard ISO 3534-11) Statistics -- Vocabulary and symbols -- Part 1: Probability and general statistical terms Corresponding Indian Standard IS 7920 (Part 1) : 2008 Statistics -- Vocabulary and symbols: Par t 1 Probability and general statistical terms (third revision) Degree of Equivalence Technically Equivalent 1) Since revised in 2006. (Continued on third cover) IS/ISO 11843-4 : 2003 Indian Standard CAPABILITY OF DETECTION PART 4 METHODOLOGY FOR COMPARING THE MINIMUM DETECTABLE VALUE WITH A GIVEN VALUE 1 Scope This part of ISO 11843 deals with the assessment of the capability of detection of a measurement method without the assumptions in ISO 11843-2 of a linear calibration curve and certain relationships between the residual standard deviation and the value of the net state variable NOTE These assumptions are often doubtful for values of the net state variable close to zero. Instead of estimating the minimum detectable value, this part of ISO 11843 provides a criterion for judging whether the minimum detectable value is less than a given level of the net state variable, and the basic experimental design for testing the conformity of this criterion. For assessment of the capability of detection, for instance as part of the validation of a measurement method, it is often sufficient to confirm that the method has a minimum detectable value that is less than a given value. 2 Normative references The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. ISO 3534-1, Statistics -- Vocabulary and symbols -- Part 1: Probability and general statistical terms ISO 3534-2:--1), Statistics -- Vocabulary and symbols -- Part 2: Applied statistics ISO 3534-3:1999, Statistics -- Vocabulary and symbols -- Part 3: Design of experiments ISO 5479:1997, Statistical interpretation of data -- Tests for departure from normal distribution ISO 5725-2:1994, Accuracy (trueness and precision) of measurement methods and results -- Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method ISO 11095:1996, Linear calibration using reference materials ISO 11843-1:1997, Capability of detection -- Part 1: Terms and definitions ISO Guide 30:1992, Terms and definitions used in connection with reference materials 1) To be published. (Revision of ISO 3534-2:1993) 1 IS/ISO 11843-4 : 2003 3 Terms and definitions For the purposes of this document, the terms and definitions given in ISO 3534 (all parts), ISO 5479, ISO 5725-2, ISO 11095, ISO 11843-1 and ISO Guide 30 apply. 4 4.1 Experimental design General The measurement method is assumed to be standardized. The same complete method shall be used for all measurements, whether of the reference states or of actual states (test samples). 4.2 Choice of reference states and reference materials The reference states shall include two values of the net state variable the value zero of the net state variable (i.e. in analytical chemistry, a sample of the blank material), and a given value, xg, which will be tested to determine whether it is greater than the minimum detectable value. The composition of the reference materials representing the reference states should be as close as possible to the composition of the material to be measured in order to satisfy the requirement that reference and test materials behave in the same way in the measuring system. 4.3 Number of replications It is assumed that the capability of detection is assessed in a separate experiment with the same number of replications for both reference states specified in 4.2. In an application of the method, measurements are performed for the reference material (representing the value zero of the net state variable) and the actual state. The number of replications used in applications of the method are usually smaller than the number of replications used in the assessment of the capability of detection of the method. The following notations are used: J is the number of replications of measurements on the reference material representing the value zero of the net state variable (blank sample) in an application of the method; K is the number of replications of measurements on the actual state (test sample) in an application of the method. N is the number of replications of measurements on each reference material (see 4.2) in assessment of the capability of detection; The value of N should preferably be at least 5. NOTE In validation of a method, the capability of detection is usually determined for J = K = 1. 5 5.1 The criterion for sufficient capability of detection Basic assumptions Basic assumptions in this part of ISO 11843 are the measurements of the response variable of all materials are assumed to be independent and normally distributed, and the reference and test materials behave in the same way in the measurement system. 2 IS/ISO 11843-4 : 2003 5.2 Critical value of the response variable When a test of the hypothesis that the net state variable of a test sample is zero is based on a comparison (in a randomized experiment) of the responses of the test sample and a sample in the basic state (blank sample known to have the net state variable equal to zero), the critical value of the response for the test sample (the mean of K measurements) is given by y c = y b + z 1- b 1 1 + J K (1) The meanings of the symbols used here and in the rest of this part of ISO 11843 are given in Annex A. When the response variable decreases with increasing level of the net state variable, the critical value of the response is given by y c = y b - z 1- b 1 1 + J K (2) where yc now is a lower limit. In this situation, the expressions g - b and y g - y b in 5.3, 5.4 and Clause 6 are changed to b - g and y b - y g respectively. 5.3 Probability of detecting a given value of the net state variable Instead of estimating the minimum detectable value of the net state variable (i.e. the value of the net state variable for which the power of the test in 5.2 has a specified value 1 - ), this part of ISO 11843 provides a criterion for the power to be greater than or equal to 1 - for a given value, xg, of the net state variable. If this criterion is satisfied, it may be concluded that the minimum detectable value is less than or equal to xg. If the standard deviation of the response for a given value xg of the net state variable is g, the criterion for the power to be greater than or equal to 1 - is given by g - b W z 1- b 1 1 + + z 1- J K 1 2 1 2 b + g J K (3) where b and g are the expected values under the actual performance conditions for the responses of the basic state and a sample with the net state variable equal to xg. NOTE Criterion (3) follows from the definition of net state variable and Figure 1 of ISO 11843-1:1997. With = , K = J and under the assumption that g W b (it is unusual for the standard deviation to decrease as the net state variable increases), the criterion is simplified to g -b 2 b 2 + g W 2 z 1- J (4) 5.4 Confirmation of the criterion for sufficient capability of detection The standard deviations and expected values of the responses in Criterion (3) are usually unknown and the fulfilment of the criterion has to be confirmed from experimental data. Thus, the expression on the left-hand side of the simplified Criterion (4) is an unknown constant, while the expression on the right-hand side is a known constant. 3 IS/ISO 11843-4 : 2003 From a validation experiment with N observations of the responses for the basic state and a sample with the net state variable equal to xg, the expression on the left-hand side of Criterion (4) is estimated by yg - yb 2 2 sb + sg (5) where the meanings for the symbols are as given in Annex A. An approximate 100(1-) % lower confidence limit (CL) for g - b ( ) 2 2 b + g is given by CL= yg - yb 2 2 sb + sg - t 1- ( ) N (6) where t1- () is the (1-)-quantile of the t-distribution with = 2(N - 1) degrees of freedom, when the hypothesis b = g is not rejected; = 2 2 ( N - 1) s b + sg 4 4 sb + sg ( ) 2 degrees of freedom according to the Welch-Satterthwaite formula, when the hypothesis b = g is rejected. If the lower confidence limit for g - b than or equal to xg is confirmed. ( ) 2 2 b + g satisfies Criterion (4), a minimum detectable value less NOTE For relatively large values of N (at least 20), it may be considered as sufficient for confirmation if either of the inequalities (3) or (4) are satisfied with the estimates y b , y g , sb and sg inserted. 6 Reporting of results from an assessment of the capability of detection From an assessment of the capability of detection, usually as part of a validation of a method, report the following: a) b) c) all relevant information about the reference materials, including the reference state value xg; the number of replicates N for each reference state; the mean values, y b and y g , and the standard deviations, sb and sg, for the responses of the basic state and the sample with the net state variable equal to xg, respectively; d) e) the chosen values of , , J and K; the left- and right-hand sides of Criterion (3) with the estimates inserted, i.e. y g - y b and z 1- s b 1 1 + + z 1- J K 1 2 1 2 sb + s g J K or, when applicable ( = , K = J and g W b), the statistic interval and its lower acceptable limit 2 z 1- f) ( yg - yb ) 2 2 sb + sg with its confidence J according to Criterion (4); the conclusion concerning the capability of detection. 4 IS/ISO 11843-4 : 2003 7 Reporting of results from an application of the method Report the observed values (responses or interpolated values of the net state variable). The fact that an observed value has been used for testing a hypothesis about the true value is no reason to discard the estimate of the true value (i.e. the observed value) and replace it by an upper limit equal to the critical value of the test or the minimum detectable value. In addition to the waste of information, it is also misleading as none of these limits may be interpreted as an upper confidence limit. Report also the applied critical value and, if possible, the minimum detectable value. 5 IS/ISO 11843-4 : 2003 Annex A (normative) Symbols used in this part of ISO 11843 J K N yc xg number of replications of measurements on the reference material representing the value zero of the net state variable (blank sample) in an application of the method number of replications of measurements on the actual state (test sample) in an application of the method number of replications of measurements on each reference material (see 4.2) in assessment of the capability of detection critical value of the response variable given value which will be tested to determine whether it is greater than the minimum detectable value expected value under actual performance conditions for responses of the basic state expected value under actual performance conditions for responses of a sample with the net state variable equal to xg standard deviation under actual performance conditions for responses of the basic state standard deviation under actual performance conditions for responses of a sample with the net state variable equal to xg observed mean response of the basic state observed mean response of a sample with the net state variable equal to xg estimate of the standard deviation of responses for the basic state estimate of the standard deviation of responses for a sample with the net state variable equal to xg (1-)-quantile of the standard normal distribution (1- )-quantile of the standard normal distribution (1- )-quantile of the t-distribution with degrees of freedom b g b g yb yg sb sg z1- z1- t1-() 6 IS/ISO 11843-4 : 2003 Annex B (informative) Example of calculation Low levels of "quickly reacting aluminium" in natural waters, expressed as mass concentration in micrograms per litre, were measured by connecting a continuous flow system to a graphite furnace atomic absorption spectrometer (see [2]). The absorbance values for five measurements of two samples representing the blank concentration xb = 0 and the net concentration xg = 0,5 µg/l are given in Table B.1. Thus, in the assessment of the method N = 5. The capability of detection is to be calculated for J = K = 1 and = = 0,05. Table B.1 -- Absorbance values for the blank concentration xb = 0 and the net concentration xg = 0,5 µg/l Net concentration of aluminium x 0 0,5 0,074 0,126 0,081 0,126 Absorbance y 0,075 0,125 0,076 0,108 0,074 0,130 The statistical analysis yields y b = 0,076 0 y g = 0,123 0 sb = 0,002 9 sg = 0,008 6 These values give yg - yb 2 2 + sg sb = 5,17 The hypothesis b = g is not rejected with an F-test at the 5 % significance level. For = 0,05 and the number of degrees of freedom = 8, then t1- (8) = 1,86, and for = 0,05 then z1- = 1,645. A 95 % lower confidence limit of g - b greater than 2 z 1- ( ) 2 2 b + g calculated according to Equation (6) is 4,34, which is J = 3,29 in Equation (4). Thus, the evaluation shows that the minimum detectable value is less than xg = 0,5 µg/l. 7 IS/ISO 11843-4 : 2003 Bibliography [1] [2] ISO 11843-2: 2000, Capability of detection -- Part 2: Methodology in the linear calibration case DANIELSSON, L.-G. and SPARÉN, A. A mechanized system for the determination of low levels of quickly reacting aluminium in natural waters. Analytica Chimica Acta, 306, 1995, pp. 173-181 8 (Continued from second cover) International Standard ISO 5479 : 1997 Statistical interpretation of data -- Tests for departure from normal distribution ISO 5725-2 : 1994 Accuracy (trueness and precision) of measurement methods and results -- Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method Corresponding Indian Standard IS 6200 (Part 3) : 1983 Statistical tests of significance: Par t 3 Tests for normality (first revision) IS 15393 (Part 2) : 2003 Accuracy (trueness and precision) of measurement methods and results: Par t 2 Basic method for the determination of repeatability and reproducibility of a standard measurement method IS/ISO 11843-1 : 1997 Capability of detection: Part 1 Terms and definitions Degree of Equivalence Technically Equivalent Identical ISO 11843-1 : 1997 Capability of detection -- Par t 1: Ter ms and definitions do The technical committee responsible for the preparation of this standard has reviewed the provisions of the following referred standards and has decided that they are acceptable for use in conjunction with this standard: International Standard ISO 3534-2 : 1993 ISO 3534-3 : 1999 ISO 11095 : 1996 ISO Guide 30 : 1992 Title Statistics -- Vocabulary and symbols -- Part 2: Statistical quality control Statistics -- Vocabulary and symbols -- Part 3: Design of experiments Linear calibration using reference materials Terms and definitions used in connection with reference materials Annex A gives a list of symbols used in this standard and is for normative reference whereas Annex B of this standard is for information only. 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